> Yes, but you get the first 1 bit for free.

Not really. :-) That's a detail of how floating-point numbers happen to be stored; it's not really relevant here. It doesn't affect the fact that IEEE 754 binary64 floats have 53 bits of *precision*, so using 31 for the integer part leaves only 22 for the fractional part, so we're 8 bits short, not 7.

(If you really want, you can subtract 30 from 52 instead of 31 from 53, but it's just a more complicated way of doing the same calculation, and doesn't change the result.)